TITLE: Is categoricity retained when reducing the language?
QUESTION [13 upvotes]: Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical.  Let $T = T’ \restriction \mathcal L$.  Is $T$ $\kappa$-categorical?
If $|\mathcal L’| = \kappa = \aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $\aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.

REPLY [5 votes]: One can also break countable categoricity when $|\mathcal{L}'|>{\aleph_0}$. This example comes from an undergraduate course I took with Malliaris. 
Let $\mathbb{P}$ be the collection of primes and let $\mathcal{P}(\mathbb{P})$ be the powerset of $\mathbb{P}$. Let $\mathcal{L'} = (+,\times,0,1;(D_{\alpha}(x))_{\alpha \in \mathcal{P}(\mathbb{P})})$ and let $T \models Th_{\mathcal{L}'}(\mathbb{N})$ with the usual interpretation of the symbols, and for each $\alpha \in \mathcal{P}(\mathbb{P})$, $\models D_{\alpha}(n)$ if and only if $n \in \alpha$. One can show that $T$ is $\aleph_0$-categorical and that the only countable model is the standard model. This follows from the fact that if one adds a single non-standard element, one must necessarily add $2^{\aleph_0}$ many elements.
However, if we let $\mathcal{L} = \{+\}$, then by Ryll-Nardzewski, we no longer have countable categoricity.